The semijoin operation is a fundamental operation of relational algebra that has been extensively used in query processing. Furthermore, semijoins have been used to formulate desirable properties of acyclic schemas; in particular, a schema is acyclic if and only if it has a full reducer, i.e., a sequence of semijoins that converts a given collection of relations to a globally consistent collection of relations. In recent years, the study of acyclicity has been extended to annotated relations, where the annotations are values from some positive commutative monoid. So far, however, it has not been known if the characterization of acyclicity in terms of full reducers extends to annotated relations. Here, we develop a theory of semijoins of annotated relations. To this effect, we first introduce the notion of a semijoin function on a monoid and then characterize the positive commutative monoids for which a semijoin function exists. After this, we introduce the notion of a full reducer for a schema on a monoid and show that the following is true for every positive commutative monoid that has the inner consistency property: a schema is acyclic if and only if it has a full reducer on that monoid.